Tricritical state and quasi-periodicity triggered by the non-linear elasticity in an Upper Convected Maxwell fluid confined between two co-oscillating cylinders about zero-mean

Abstract

The effects of harmonically co-oscillating the inner and outer cylinders about zero mean rotation in a Taylor–Couette flow are examined numerically using Floquet theory, for the case where the fluid confined between the cylinders obeys the upper convected Maxwell model. Although stability diagrams and mode competition involved in the system were clearly elucidated recently by Hayani Choujaa et al. (2021) in weakly elastic fluids, attention is focused, in this paper, on the dynamic of the system at higher elasticity with emphasis on the nature of the primary bifurcation. In this framework, we are dealing with pure inertio-elastic parametric resonant instabilities where the elastic and inertial mechanisms are considered of the same order of magnitude. It turns out, on the one hand, that the fluid elasticity gives rise, at the onset of instability, to the appearance of a family of new harmonic modes having different axial wavelengths and breaking the spatio-temporal symmetry of the base flow: invariance in the axial direction generating the $O\left(2\right)$ symmetry group and a half-period-reflection symmetry in the azimuthal direction generating a spatio-temporal $Z_2$ symmetry group. On the other hand, new quasi-periodic flow emerging in the high frequency limit and other interesting bifurcation phenomena including bi and tricritical states are also among the features induced by the fluid elasticity. Lastly, and in comparison with the Newtonian configuration of this system, the fluid elasticity leads to a total suppression of the non-reversing flow besides emergence of instabilities with lower wavelengths. Such a comparison provides insights into the dynamics of elastic hoop stresses in altering the flow reversal in modulated Taylor–Couette flow.